Birkhoff [1]} discovered that a substantial amount of mathematics can be done at the level of generality of an equational theory. Given an equational theory \((\Sigma ,E)\) , a model is a set together with an interpretation of the function symbols \(\Sigma \)  that satisfies the equations \(E\) . A monoid is nothing but a model of the equational theory of monoids, a group is a model of the equational theory of groups, and so on. The semantics of an equational theory, i.e. its class of models, is called a variety. Birkhoff showed that certain results (e.g. the so-called isomorphism theorems, existence of free models) can be derived uniformly for generic varieties, independent of the equational theory at hand. Most spectacularly, a class of sets-with-structure can be determined to be a variety through purely structural means; this is often referred to as Birkhoff's Variety Theorem or the HSP Theorem.
